3 LOGARITHMS AND EXPONENTIALS 85

We first prove a lemma:
(4.3.1.1) For any x > 0, there is an integer m (positive or negative) such

Suppose first x ^ 1. The sequence (an) is strictly increasing. If we had
an < x for all integers n>Q, then (by (4.2.1 ) and (3 .1 5.4)) b = lim an = sup a11

n-*oo n

would be finite, > 1 and ^x; but we can write b = lim an + i = a -Urn a11

n-+oo jj~*oo

by (4.1.2), hence b = ab, which contradicts the assumption a > 1. Therefore
there is an integer n such that x > a"; take m + 1 as the smallest of these inte-
gers. If on the contrary 0 < x < 1, then x~l > 1, and if am < Jt"1 ^am + \
we have a'(m+1) < x < <r(m+1)+1 (2.2.13).

Suppose there exists a function / having the properties listed in (4.3.1);
then /is a homomorphism of the multiplicative group R* into the additive
group R, and therefore we must have /(I) = 0,/(.xn) = n -f(x) for any x > 0
and any integer n (positive or negative), and in particular f(an) = n. More-
over, if am ^ x" < am+\ we must have f(am} </(*") </(<zm+1), in other
words m ^ n -f(x) < m + 1, hence m/n ^f(x) and \f(x) — m/n\ < l/n.
This shows that if we denote by Ax the set of rational numbers m/n (m positive
or negative, n^l) such that am sŁ x", (note that am < xn and amq < xnq,
where q is an integer >0, are equivalent relations (2.2.13)), we must have
f(x) = sup Ax , which shows / is unique.

To prove the existence of /, it remains to prove that the mapping /:
^-j-supAjc verifies all our conditions. Let x and y be any two elements
of R* ; for any integer n ^ 1, let m, m' be such that am < x" < am+l and
am' < / ^ am' + 1 ; from these relations it follows that am+m> < (xy)n ^am+m' + 2;
hence we have

m „, v m + 1 m' „, , m' + 1

n n n

m + m' m + m'4-2

- </(*)>) ^ - ,

and also

m + m' + 2

—j^-</(*)+/00«-

We conclude that
and as ra is arbitrary,rs; if, to each subset X